Question: Simplify and expand the following expression: $ \dfrac{4}{q - 9}- \dfrac{1}{q + 4}- \dfrac{4}{q^2 - 5q - 36} $
Solution: First find a common denominator by finding the least common multiple of the denominators. Try factoring the denominators. We can factor the quadratic in the third term: $ \dfrac{4}{q^2 - 5q - 36} = \dfrac{4}{(q - 9)(q + 4)}$ Now we have: $ \dfrac{4}{q - 9}- \dfrac{1}{q + 4}- \dfrac{4}{(q - 9)(q + 4)} $ The least common multiple of the denominators is: $ (q - 9)(q + 4)$ In order to get the first term over $(q - 9)(q + 4)$ , multiply by $\dfrac{q + 4}{q + 4}$ $ \dfrac{4}{q - 9} \times \dfrac{q + 4}{q + 4} = \dfrac{4(q + 4)}{(q - 9)(q + 4)} $ In order to get the second term over $(q - 9)(q + 4)$ , multiply by $\dfrac{q - 9}{q - 9}$ $ \dfrac{1}{q + 4} \times \dfrac{q - 9}{q - 9} = \dfrac{q - 9}{(q - 9)(q + 4)} $ Now we have: $ \dfrac{4(q + 4)}{(q - 9)(q + 4)} - \dfrac{q - 9}{(q - 9)(q + 4)} - \dfrac{4}{(q - 9)(q + 4)} $ $ = \dfrac{ 4(q + 4) - (q - 9) - 4} {(q - 9)(q + 4)} $ Expand: $ = \dfrac{4q + 16 - q + 9 - 4}{q^2 - 5q - 36} $ $ = \dfrac{3q + 21}{q^2 - 5q - 36}$